Minimum Aberration Designs Under Orthogonal And Baseline Parameterization

The design and analysis of factorial experiments is a significant research direction within the field of experimental design.Fractional factorial designs have gained widespread application and attention due to their efficiency and cost-effectiveness.The comparison and selection of fractional factorial designs have consistently been a crucial research topic in the field of factorial design.The minimum aberration(MA)criterion is one of the most popular criteria for assessing the goodness of fractional factorial designs.The linear model is commonly employed in the analysis of experimental data,and typically involves two types of parameter constraints:zero-sum constraint and baseline constraint.In factorial design studies,the most commonly used to define factor effects is based on a set of mutually orthogonal treatment contrasts.This model,based on zero-sum constraint,is referred to as an orthogonal parameterization model.The model based on baseline constraint is known as a baseline parameterization model.Under baseline parameterization,factor effects are measured by the impact on the response when one or more factors change levels while keeping the other factors at their baseline levels.Optimal designs under these two parameterization models may differ.Existing literature on baseline designs under the MA criterion predominantly focuses on two-level designs,with limited research on high-level and mixed-level designs.This dissertation is devoted to the study of the selection and construction of MA designs under two parameterization models.Chapter 1 outlines the background and preliminary knowledge for the study.Chapter 2,by deriving the bias contributions of non-negligible interaction effects on the estimation of main effects,explores three-level baseline designs under the MA criterion.It introduces the MA criterion under the baseline parameterization for comparing and selecting three-level designs.Considering that the interchangeability of baseline and test levels in baseline designs renders the definition of design isomorphism under orthogonal parameterization inapplicable,this chapter establishes a definition for isomorphism of three-level baseline designs.For optimal design selection,the study employs a complete search algorithm using all non-isomorphic three-level orthogonal arrays as the initial designs,resulting in optimal designs for 18-run and 27-run.Two incomplete search methods that produce nearly optimal designs are proposed for situations where a complete search is impracticable.Chapter 3 extends the research from Chapter 2 to high-level baseline designs.It introduces the definition of isomorphism for high-level baseline designs and proposes the MA criterion applicable to any s-level baseline design(where s≥2).Utilizing non-isomorphic orthogonal arrays with s-level as initial designs,the chapter presents(nearly)optimal designs for the number of levels from 2 to 5 and runs from 8 to 24.Since it is challenging to ensure all factors have the same levels in most experiments,mixed-level experimental designs find wide application.Chapters 4 and 5 investigate mixed-level designs under two parameterizations.Chapter 4 extends the MA criterion under the baseline parameterization to mixed-level designs,using nonisomorphic mixed-level orthogonal arrays as initial designs.It proposes a complete search algorithm and provides(nearly)optimal baseline designs for 8-run to 20-run.The results for three-level and s-level designs are special cases of this chapter.Considering the non-orthogonality of baseline designs,which renders the decomposition of sums of squares in the analysis of variance inapplicable,the chapter details the analysis method for baseline experimental data through an example of a three-factor experiment.Chapter 5 examines mixed-level fractional factorial split-plot designs under the MA criterion for orthogonal parameterization.Due to the two-stage randomization in the fractional factorial split-plot designs,errors generated by WP randomization are larger than those from SP randomization.Thus,the two kinds of factors cannot be treated equally.The chapter investigates mixed-level fractional factorial splitplot designs with a four-level factor when the experimenter has prior knowledge that WP factors are more important than SP factors.It introduces the combination minimum aberration of type WP criterion when WP factors are more important than SP factors,constructs optimal 2(n1+n2)-(k1+k2)4w1 designs for small values of k1 and k2,and provides optimal design tables for 8-run,16-run,32-run and 64-run.Chapter 6 concludes the dissertation by summarizing the research and providing prospects for future studies.